Operationally-Motivated Uncertainty Relations for Joint Measurability and the Error-Disturbance Tradeoff

Abstract

We derive new Heisenberg-type uncertainty relations for both joint measurability and the error-disturbance tradeoff for arbitrary observables of finite-dimensional systems. The relations are formulated in terms of a directly operational quantity, namely the probability of distinguishing the actual operation of a device from its hypothetical ideal, by any possible testing procedure whatsoever. Moreover, they may be directly applied in information processing settings, for example to infer that devices which can faithfully transmit information regarding one observable do not leak any information about conjugate observables to the environment. Though intuitively apparent from Heisenberg's original arguments, only more limited versions of this statement have previously been formalized.